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Large deviation principle for epidemic models

Part of: Stochastic analysis Limit theorems Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  15 September 2017

Etienne Pardoux  and
Brice Samegni-Kepgnou
Etienne Pardoux*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
Brice Samegni-Kepgnou*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
*
* Postal address: CNRS UMR 7373 (I2M), Aix-Marseille Université, CMI, 39 rue F. Joliot Curie, F-13453 Marseille, France.
* Postal address: CNRS UMR 7373 (I2M), Aix-Marseille Université, CMI, 39 rue F. Joliot Curie, F-13453 Marseille, France.
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Abstract

We consider a general class of epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes and we show the large deviation principle for such models. We generalise the approach followed by Dolgoarshinnykh (2009) in the case of the SIR epidemic model. Thanks to an additional assumption which is satisfied in many examples, we simplify the recent work of Kratz and Pardoux (2017).

Keywords

Poisson processlarge deviation principlelaw of large numbers

MSC classification

Primary: 60F10: Large deviations 60H10: Stochastic ordinary differential equations 60J75: Jump processes 92C60: Medical epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 905 - 920
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. CrossRefGoogle Scholar
[2] Britton, T. and Pardoux, E. (2017). Stochastic epidemics in a homogeneous community. In preparation. Google Scholar
[3] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Springer, Berlin. Google Scholar
[4] Dolgoarshinnykh, R. (2009). Sample path large deviations for SIRS epidemic processes. Submitted. Google Scholar
[5] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York. CrossRefGoogle Scholar
[6] Dupuis, P., Ellis, R. S. and Weiss, A. (1991). Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Prob. 19, 1280–1297. Google Scholar
[7] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. John Wiley, Hoboken, NJ. Google Scholar
[8] Feng, J. and Kurtz, T. G. (2006). Large Deviations for Stochastic Processes (Math. Surveys Monogr. 131). American Mathematical Society, Providence, RI. Google Scholar
[9] Fierro, R. (1996). Large-sample analysis for a stochastic epidemic model and its parameter estimators. J. Math. Biol. 34, 843–856. CrossRefGoogle ScholarPubMed
[10] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd edn. Springer, Heidelberg. CrossRefGoogle Scholar
[11] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[12] Kratz, P. and Pardoux, E. (2017). Large Deviation for Infections Diseases. In Séminaire de Probabilités XLIX (Lecture Notes Math.), Springer (to appear). Google Scholar
[13] Kribs-Zaleta, C. M. and Velasco-Hernández, J. X. (2000). A simple vaccination model with multiple endemic states. Math. Biosci. 164, 183–201. CrossRefGoogle ScholarPubMed
[14] Kurtz, T. G. (1978). Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6, 223–240. CrossRefGoogle Scholar
[15] Léonard, C. (1990). Some epidemic systems are long range interacting particle systems. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86), Springer, Berlin, pp. 170–183. CrossRefGoogle Scholar
[16] Safan, M., Heesterbeek, H. and Dietz, K. (2006). The minimum effort required to eradicate infections in models with backward bifurcation. J. Math. Biol. 53, 703–718. Google Scholar
[17] Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis. Chapman & Hall, London. Google Scholar
[18] Shwartz, A. and Weiss, A. (2005). Large deviations with diminishing rates. Math. Operat. Res. 30, 281–310. Google Scholar
[19] Sokol, A. and Hansen, N. R. (2015). Exponential martingales and change of measure for counting processes. Stoch. Anal. Appl. 33, 823–843. Google Scholar